240 XIII INTEGRATION (cf. Problem 1). For the general case, consider the functions sup(w, f<pKn), where (Kn) is an increasing sequence of compact sets whose union is X.) 20. Let U be a continuous endomorphism of the space Li(X, /it) satisfying the hypotheses of Section 13.11, Problem 17, so that U extends to an endomorphism of each of the spaces L£(X, JLC), 1 <>p <; -f oo (he. cit.). For each function fe && , put R«(/) = - (/+#•/+••' + U"'1 •/), n R*(/) = sup |Rk(/)|, l^k^n R*(/) = supR*(/). n (a) Show that if 1< p < + oo and / e -2=$ , then R*(/) e -2* and p,. P— I (Use Problem 19 above, and Section 13.11, Problem 17(c).) (b) For each function/ e J§?£, let P -/be the limit in Lj| of the sequence ((Rn(/))~) (Section 12.15, Problem 12(c)). We have £/P = P(7 = P. Show that the sequence (Rn(/)) converges almost everywhere to P -/(Dunford-Schwartz ergodic theorem). (Put S(/) = lim sup Rn(/), rt-»OO = liminfRM(/). Observe that !(/) g S(/) ^ R*(/), and that S(Rm(/) -P-/) = S(/)-P-/ and I(Rm(/) -P */) = K/) -P -/for all m £ 1, and use (a) with Rm(/) -P • /in place of/) (c) Show that N^P •/) £ NI(/) for all / e && n J2?ff , and hence that P extends to a contraction on the space LR. Deduce that the sequence (R»(/)) converges almost everywhere to/, for each function /e jfifj. (Put L(/) = limsup|Rw(/)-P-/|, n-*oo observe that L(/) ^ R*(/-^) + P • \f-g\ for all ^ e Se\ n jgff , and make use of Section 13.11, Problem 7(d).) 21. The notation is that of Section 13.17, Problem 7. For each a > 0 and each function g e &}OCtC(y, v), let AM denote the set of points y e Y such that \g(y)\ > a. If p e [1, + oo [, the endomorphism U is said to be of weak type (p,p) if there exists a constant C > 0 such that, for each fc-integrable step function / and each a > 0, we have and that